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Are there axioms that can be added to ZF to get a system, call it ZF+, that will be weaker than ZFC, but satisfy a stronger version of the absoluteness theorem, with regards to removing choice from proofs, in the sense indicated in the title? More generally, since we can do the same for ZF + ◊, and anything else that holds in the L, can we just as easily replace choice with something else; or say something about how these middle systems' non-ZF axioms relate to how far up the AH the result goes? Final question: something like the above, but systems between ZF and ZF + V=L; and, in that case, would the degree up the hierarchy be related to "how close" the system was to V = L? (I am having a slightly stupid day and feel like this question may not make perfect sense, please excuse me if I am saying something confused).Phoenixia1177 (talk) 08:04, 6 December 2014 (UTC)
For those that reject/dislike choice on philosophical grounds, is there anything compelling about choice holding in L? I realize that L isn't "constructive", but for a certain type of dislike of choice, it seems that this would be persuasive on some level. Then again, I don't, to be perfectly honest, really "get" objections to choice that fall under this nature - intellectually, yes; but philosophically, no.Phoenixia1177 (talk) 08:20, 6 December 2014 (UTC)
I think it would be quite convenient for pi to be an integer (like 3). Is there any reason that this obviously would not have been coincidentally possible? 212.96.61.236 (talk) 14:29, 6 December 2014 (UTC)
Guys, I mean that you have to calculate pi to know what it is (or you have to try to measure it.) When you start your calculation you don't know what you'll get. I know that for thousands of years we've gotten something like 3.14... My question is a priori, when you start calculating, how could you already know it won't be a whole number? What I'm asking is very similar to the question of the irrationality of pi. (Since all integers are rational, in fact the irrationality of pi is a fine answer to my question. These are proofs that pi is irrational and by extension not an integer. But my question is simpler than proving that it's not rational: give me a sentence or two that, without having to calculate pi, intuitively proves that it doesn't make sense for it to be an integer. (i.e. give me an intuitive, weaker version of the proof that it's irrational - produce an easy intuitive proof that it isn't integral.) 212.96.61.236 (talk) 17:45, 6 December 2014 (UTC)
When uploading encrypted files you always get a bit of an existential crisis: Is it really encrypted? So I thought to create an automatic check, but to my dismay the standard randomness test, chi-square test, at least as implemented by ent, can't tell compressed data from encrypted with a good enough reliability. I ran ent against a bunch of files and their encrypted versions, and a good number of compressed files came out as more random than encrypted files (the offending cases are in bold):
Chi-square | File |
---|---|
217.221764 | test-3.csv.gpg |
225.000781 | test-2.csv.gpg |
234.114763 | test-2.csv.xz.gpg |
235.953133 | test-4.csv.xz.gpg |
237.257606 | test-1.csv.bz2.gpg |
237.992235 | test-4.csv.gz.gpg |
241.935213 | test-5.csv.xz |
241.987507 | test-1.csv.xz |
242.404189 | test-1.csv.gz.gpg |
247.759906 | test-3.csv.bz2.gpg |
250.514585 | test-3.csv.gz.gpg |
251.933691 | test-1.csv.xz.gpg |
256.686651 | test-4.csv.xz |
257.086558 | test-2.csv.bz2.gpg |
257.571442 | test-3.csv.xz.gpg |
258.278380 | test-4.csv.bz2.gpg |
259.446203 | test-5.csv.gz.gpg |
262.789336 | test-1.csv.gpg |
263.092097 | test-3.csv.xz |
263.181771 | test-5.csv.gpg |
265.921566 | test-2.csv.xz |
273.224329 | test-5.csv.xz.gpg |
273.278069 | test-5.csv.bz2.gpg |
276.475462 | test-2.csv.gz.gpg |
291.546158 | test-4.csv.gpg |
1966.343430 | test-1.csv.gz |
6702.253283 | test-1.csv.bz2 |
12985.448647 | test-2.csv.gz |
15104.200254 | test-5.csv.gz |
16030.483501 | test-5.csv.bz2 |
28917.750924 | test-4.csv.gz |
33106.068205 | test-2.csv.bz2 |
35213.209600 | test-4.csv.bz2 |
47059.290959 | test-3.csv.gz |
64590.806830 | test-3.csv.bz2 |
9864569.330480 | test-1.csv |
75136408.248885 | test-5.csv |
100468120.894992 | test-2.csv |
167571241.918255 | test-4.csv |
210238442.386372 | test-3.csv |
As you can see xz compression is pretty indistinguishable from encrypted data. Limiting the test to the first 32 bytes of each file, where header data would reside, clearly separates the .xz files:
Chi-square | File |
---|---|
240.000000 | test-4.csv.gz.gpg |
256.000000 | test-1.csv.xz.gpg |
256.000000 | test-2.csv.gz.gpg |
256.000000 | test-2.csv.xz.gpg |
256.000000 | test-3.csv.bz2.gpg |
256.000000 | test-4.csv.gpg |
272.000000 | test-1.csv.gpg |
272.000000 | test-1.csv.gz |
272.000000 | test-2.csv.bz2.gpg |
272.000000 | test-3.csv.gpg |
272.000000 | test-3.csv.xz.gpg |
272.000000 | test-4.csv.xz.gpg |
272.000000 | test-5.csv.bz2 |
272.000000 | test-5.csv.gz.gpg |
288.000000 | test-1.csv.bz2.gpg |
288.000000 | test-1.csv.gz.gpg |
288.000000 | test-2.csv.gpg |
288.000000 | test-3.csv.bz2 |
288.000000 | test-3.csv.gz |
288.000000 | test-3.csv.gz.gpg |
288.000000 | test-4.csv.bz2.gpg |
288.000000 | test-4.csv.gz |
288.000000 | test-5.csv.bz2.gpg |
288.000000 | test-5.csv.gpg |
288.000000 | test-5.csv.gz |
304.000000 | test-1.csv.bz2 |
304.000000 | test-2.csv.gz |
304.000000 | test-4.csv.bz2 |
320.000000 | test-2.csv.bz2 |
320.000000 | test-5.csv.xz.gpg |
560.000000 | test-5.csv |
576.000000 | test-1.csv |
576.000000 | test-3.csv.xz |
576.000000 | test-4.csv.xz |
576.000000 | test-5.csv.xz |
592.000000 | test-1.csv.xz |
688.000000 | test-2.csv.xz |
704.000000 | test-3.csv |
736.000000 | test-4.csv |
752.000000 | test-2.csv |
If combined with the values for full files, this is great for this limited set of file formats, considering for a chi-square distribution of 255 degrees of freedom (256 values in a byte) a value of more than 500 occurs with a probability around 10^-17. But if you don't include any special knowledge about where the non-random data is, say if you check every 32 byte block, at %50 false negative probability you could only upload roughly one exabyte (about a million raw hard disk images), which is less than stellar for whoever gets hit by that.
Now I'm sure if you play with the details you could might gain a few orders of magnitude (e.g. you get 6 if you raise the threshold to 550) but there could also be plenty of hiding weaknesses (e.g. I haven't even bothered to figure out a false positive rate (not that I'm looking for a silver bullet)). What I'd really like to ask is if you have any ideas that would decisively improve on this. Do you know a test other than chi-square that would perform better, or can you think of some modification to the chi-square scheme? One idea to square the false negative rate is to generate a somehow related-in-randomness-yet-independent-in-chi-square block if a block fails, and then also test the related block, but then I've got no idea if a thing like that exists. --Swedmann (talk) 20:25, 6 December 2014 (UTC)